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Symmetry

32 33 BrAnchinG systems patterns of distribution Branched networks can be thought of as having a real existence, like those of trees, rivers, etc., or simply as mental concepts that exist independently of any physical representation. In the latter case, fairly complex systems can be generated from quite simple rules lower opp.. One of the more fascinating aspects of branching is that similar habits can be expressed in entirely different settings there are, for instance, branching hierarchies in lightning strikes that closely resemble those in river systems. There may even be a close correspondence between formations that disperse and those that concentrate below. In either case, functioning branching systems involve the efficient distribution of energy in one form or anotherthey are the simplest way to connect every part of a given area using the shortest overall distance or least work. The hidden symmetries operating within branching formations concern the rates and ratios of bifurcation. In a simple progression, for instance, three streamlets may feed into a stream, three streams into each tributary, and finally, three tributaries into a river. This sort of progression is, in fact, a common pattern, found not only in rivers and plants, but in animal vascular systems. Although the rules that determine branching in nature tend to be more involved than this, nevertheless, relatively simple algorithms may create highly complex forms. 33 The general characteristics of all branching patterns, whether of river systems, electric discharges or those of biological systems, is that they radiate out or converge in, and that any branches of a particular size are always outnumbered by those of the next smaller dimension
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